On the developments of Darcy's law to include inertial and slip effects

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On the developments of Darcy’s law to include inertial and slip effects

Didier Lasseux, Francisco Valdés-Parada

To cite this version:

Didier Lasseux, Francisco Valdés-Parada. On the developments of Darcy’s law to include inertial and slip effects. Comptes Rendus Mécanique, Elsevier, 2017, 345 (9), pp.660-669.

�10.1016/j.crme.2017.06.005�. �hal-03140868�

C. R. Mecanique 345 (2017) pp. 660–669 - DOI : 10.1016/j.crme.2017.06.005

A century of fluid mechanics: 1870–1970 / Un siècle de mécanique des fluides : 1870–1970

On the developments of Darcy’s law to include inertial and slip effects

Didier Lasseux

^a,∗

, Francisco J. Valdés-Parada

^b

aCNRS,UniversitédeBordeaux,I2M–UMR5295,EsplanadedesArts-et-Métiers,33405Talencecedex,France

bUniversidadAutónomaMetropolitana-Iztapalapa,DepartamentodeIngenieríadeProcesoseHidráulica,Av.SanRafaelAtlixco186, 09340 Mexico,D.F.,Mexico

a b s t ra c t

Keywords:

Porousmedia Inertialflow Slipflow Darcy’slaw

Forchheimercorrection Klinkenbergcorrection

TheempiricalDarcylaw describing flowinporous media, whoseconvincing theoretical justification was proposed almost 130 years after its original publication in 1856, has however been extended to account for particular flow conditions. This article reviews historicaldevelopmentsaimedatincludinginertialandslipeffects(respectively,whenthe ReynoldsandKnudsennumbersarenotexceedinglysmallcomparedtounity).Despitethe earlyempiricalextensionstoincludeinertiaandslipeffects,itisstrikingtoobservethat clearformalderivationsofphysicalmodelstoaccountfortheseeffectswerereportedonly recently.

1. Introduction

Flowsinporousmediaareofinterestinnumerousapplicationsrangingfromhydrology,hydrocarbonrecovery,gasand nuclear wastestorage, todryingof wood, transferinfoodproducts orin livingtissues tocite butafew.The mainchar- acteristic ofthisparticular domainoffluidmechanics liesinthe(sometimesextreme)complexityofthe geometryofthe channels where the flow takesplace. Additionally, in manysituations, this geometryis unknown in its very details and mayvaryovermoreorlesslongdistancescharacteristicsofheterogeneities.Withinthiscontext,thephysicaldescriptionof theflowinsuchmaterialsmayappeartobeatremendouschallenge.¹ Thiscertainlyexplainswhyempiricismremainedso strongandlastedlongerthaninmanyotherfieldsoffluidmechanics.Inmanysituations,theinterestisnotinthedetailsof theflowwithintheporesbutratherintheflux-to-forcegoverninglawsatlengthscalesincludingalargenumbersofpores, althoughcomprehensiveanalysesattheporescaleremainthecornerstoneinanyprogresstowardsthederivationofgov- erninglawsatlargerscales. Clearly,activeresearchinthedescriptionoftransferinporousmaterials wastriggeredby the publicationofDarcy’slawinthemiddleofthe19thcenturyandtheemergenceofakeymacroscopicphysicalcharacteristic ofaporousmedium,namelytheabilityofafluidtoflowthroughit,i.e.itspermeability.

*

Correspondingauthor.

E-mailaddresses: didier.lasseux@u-bordeaux.fr(D. Lasseux),iqfv@xanum.uam.mx (F.J. Valdés-Parada).

1 Theone-phaseslowflowisprobablyoneofthesimplestmechanismonecanthinkaboutandthereisalotofothertremendouslymorecomplex physicalprocessesinporousmediaofrelevancefrombothscientificandindustrialpointsofview,includingmultiphaseflows,compressibleflows,phase change,deformableporousmedia,reactionsinmulticomponentsystems,etc.

1.1. Darcy’slawasanempiricalbasis

Ever since its empiricalformulation in 1856, Darcy’s law [1] has been a hallmark in modeling momentum transport throughporous media.Inthisclassical publication,thereis asection entitledDéterminationdesloisd’écoulementdel’eauà traverslesable,dedicatedtothestudyofwaterflows throughabedofsandwherethefollowingrelationisproposed(see page 594in [1]):

q

=

^ks

e

(

h

+

^e

)

(1)

whereqisthevolumetricflowrate,s isthecrosssectionofthesandbed, eisthebedwidth,h isthepressure(orhead) differencebetweenthesurfaceandthebaseofthesandbed, andfinallyk was proposedasacoefficientthat dependson thepermeabilityofthebed²andonthepropertiesofthefluid.ForanexcellentreviewabouttheoriginofDarcy’slaw,the interested readerisreferredtotheworkby Zerner[3].The useofthissimplerelationrequiresthattheonlyresistanceto theflowthroughtheporousmediumisduetoviscousstressesinducedbyanisothermal,creeping(orlaminar)steadyflow ofa Newtonian fluid within an inert, rigidandhomogeneous porous medium.However, the lackof a rigorousupscaling techniquepreventedaformalderivationofthisequationuntilthelate1960s,asitwillbedetailedlater.

Foraverylongperiodoftime–aroundfiftyyears–thislawhasbeenessentiallyverifiedexperimentallyinits global form,butwasnotconsideredinalocaldifferentialformnorderivedonatheoreticalbasis.Onefindsadifferentialexpres- sionintheanalysisoftheflowsinaquifersbyJ.Boussinesq[4]asaresultofananalogywithheattransferinacontinuum.

This workalso reportsan extension ofthe flow-rate-to-head-gradient relationshipto non-homogeneous media. A formal derivation ofa1D localexpression ofthislawobtainedfromthe solutionto theStokes equationfor aflow parallelto a regular array of infiniteparallel cylinders (sufficiently apartfrom each other,i.e. forrelatively large porosities) is dueto Emerslebenin 1925[5].A derivationmainly basedon dimensionalanalysiswas later proposed by Muskat andBotsetin 1931 [2]fora compressible flow inwhich the pressure difference is recognized to be replaced by the difference ofthe squaresofthepressures.

Substantial literature will then appear during the 1950s, inwhich many differentapproaches to demonstrateDarcy’s lawwillbe tested(seeforinstance[6]andreferencestherein).Althoughthesearticleshelpedprogressing intotheunder- standingoftheapplicabilityofDarcy’slaw,almostallofthemreliedonanalogies, hypothesesorpostulatesthat leftthem incomplete. The first extension to three-dimensions and to non-isotropic materials was reported by Hall [7], who intro- ducedapermeabilitytensor,whichisalsobasedonsomepre-requisites (seeinparticularEq.(17)thereinandthewaythe permeabilityisidentified).

DespitethelackofformalderivationofDarcy’slaw,whichcanbeexpressedfora1Dflowinthex-directionas[8]

q

= −

^{K s}

μ

∂

^pβ

^β

∂

x (2)

themeaningofthepermeabilityanditsrelationshiptotheunderlyingporestructurefocusedclosedattention.Intheabove expression,K isthepermeabilityhavingunitsofm² and

μ

thefluidviscosity.Inaddition,^pβ^βistheintrinsically-averaged pressureintheporousmedium,definedas:

^pβ

^β

=

¹ V_β

Vβ

p_βdV (3)

Here,Vβ isthedomain(ofvolume Vβ) occupiedbythefluidphase β withinarepresentativeaveraging domain(orREV) (seeFig. 1),andpβ isthepore-scalepressure.

AnearlyestimateofK wasinspiredbyananalysisduetoBlakein1922[9]offlowoverpackingsofgrainsofdifferent shapesandacomparisontoflowsincapillarytubesthatresultedinthefollowingestimate

K

=

¹ k₀S²₀

ε

³

(

1

− ε )

^{2 (4)}

where

ε

istheporousmediumporosity,S0 denotesthespecificsurfaceoftheparticlesanditisdefinedastheratioofthe area oftheparticletoitsvolume. Thecoefficient S0,relatedtothe effectiveparticlediameter,dp,was identifiedfroman analogywithsphericalparticlesby

S0

=

⁶ dp

(5)

2 In[1],H. Darcyindicatedthatk“dependsonthepermeabilityofthesandlayer”.Infact,kisthehydraulicconductivity,havingtheunitofalengthper unittime.Theintrinsicpermeabilityasaphysicalquantity,denotedbyK(orKintensorialform)inthepresentarticle,appearedlaterintheliterature.It seemsthatM.Muskat(seeforinstance[2])wasthefirstwhousedthiscoefficient.

Fig. 1.Sketch of a porous medium including an averaging domain and the phases involved.

Finally, k0 isan adjustablecoefficient thatwas later known asthe Kozeny[10] coefficient. Carman[11] suggestedtaking k0=^{5,sothatequation(4)cannowbeexpressedas}

K

=

^d

2p

ε

³

180

(

1

− ε )

^{2 (6)}

whichisusuallyknownastheKozeny–Carmanequationandwhichisoftenused,sometimesabusively,topredictthevalue ofpermeability.Certainly,aprecisecorrelationforanytypeofporestructureisoutofreach.

1.2. TheoreticalfoundationofDarcy’slaw

The lack ofconvincing formal derivations ofDarcy’s lawthat lastedover more thana century is obviouslyrelated to the lackofclearupscaling methodsallowing oneto obtainthemacroscopic conservationequationsformtheir microscopic (i.e.pore-scale) counterparts,aspointedout byZerner [3]. Thesemethodsemerged inthe1970sanda firstattemptwas proposedbyS. Whitakerin1966[12],whoclearlyobtainedageneralizationofDarcy’slawinthefollowingvectorialform:

^vβ

= −

^K

μ ^·

∇

^pβ

^β

− ρ

g

(7) where

ρ

isthefluiddensity,gisthegravityacceleration,Kisthepermeabilitytensorand^vβ^{theseepagevelocity,which} isdefinedasthesuperficialaverage:

^vβ

=

¹ V

Vβ

v_βdV (8)

with V beingthevolume oftheREV andv_β the pore-scalevelocity vector. The sameresultwas achievedexactlyin the same period by C. Marle[13]. However, inthese articles,no clearstructurallink (i.e.a closure) is provided betweenthe micro- andthemacroscaleallowing onetoinferthedependenceofthepermeabilityuponthepore-scalegeometry.Itwas not earlierthaninthe middleofthe1980sthat aclosed rigorousformwas achievedbyWhitaker[14] usingthe volume averaging method[15],whichincluded anintrinsicclosureschemeallowingone topredict thevaluesofthecomponents ofthe permeabilitytensor K.Some derivations ofDarcy’slawhaveusedotherupscaling techniquessuch ashomogeniza- tion[16].Theseandothertechniquesdepartfromthegoverningequationsattheporescaleand,aftertheapplicationofan averaging operator(suchastheone sketchedinFig. 1)andmanymathematicalsteps,leadtoanupscaledmodelinterms ofeffective-mediumcoefficientsthatcapturetheessential(i.e.non-redundant)informationfromtheporescale.Inthisway, thepermeabilitytensorcanbeviewedasasignatureoftheporousmediumtopologyatascalethatislargerthanthepore scale.

Over the past century,there have been some modifications brought to Darcy’s law that extended its applicability to more complicatedtransport processesthan thoseoriginally considered by Darcy.Among the extensionsto Darcy’slaw,a non-exhaustivelistshouldinclude:1) Forchheimer’smodificationtoallowforthestudyofnon-creepingflowregimes[17];

2) Brinkman’s correction to include macroscopic viscous stress by introducing an effective viscosity coefficient [18];

3) Klinkenberg’s modificationof the permeabilitytensor to study gasslip flows in porous media [19]; 4) application to heterogeneous media by means oflarge-scale volumeaveraging [20]; 5) non-isothermalflow of non-Newtonianfluidsin porousmedia(cf.[21]forinstance).

Inwhatfollows,thefocusislaidupon twoextensionstoDarcy’slawthatareofmajorimportance,namelytheinertial one-phaseflowandthegasslipflowinhomogeneousporousmedia.Ouraimistocarefullyreviewtheseextensionsanddraw someconclusionsandperspectivesonthesesubjects.

2. Inertialone-phaseflowinporousmedia

Theanalogywithflows inductshasbeenwidely usedinthederivationofempiricalflowmodelsinporousmediaand mighthavebeenasourceofinspirationforH.Darcytoobtainthefiltrationlawhereportedinhisbook[1].Asmentioned inaremarkableanalysisbyZerner[3],H. DarcydedicatedspecificexperimentstoverifyPoiseuille’slawinthecontext of slowflow.Oneofthemajormotivationswashisquestioningofthevalidityoftherelationshipbetweenthe“pressuredrop”

P andtheaveragevelocityuinatubeoflength L,whichwasthenofcommonuse,i.e.

P

L

=

^au

+

^{bu2 (9)}

arelationshipmainlyduetoduBuatandGaspardRichedeProny(see[3]),withaandb beingcoefficientsthat hadtobe determinedexperimentally.

2.1. Forchheimer’scorrectiontoDarcy’slaw

ItisstrikingtoobservethattheformofEq.(9)correspondstotherelationshipproposedbyForchheimer[17]toaccount for“rapid”flowsinporousmediawiththeclassicalanalogyonu takenasthefiltrationorseepagevelocityin1D:

P L

= μ

Ku

+ ρ β

u² (10)

where

ρ

isthefluiddensityandβ istheForchheimercoefficient,alsocalledthecoefficientofinertialresistanceorinertial resistancefactor.

ThequadraticcorrectionintroducedbyForchheimerabouthalfacenturyaftertheempiricalvalidationandpublicationof Darcy’slawwasobviouslyinspiredfromEq.(9),despitethisremarkabletimelapsebetweenForchheimer’spublicationand that ofDarcy.In1931, MuskatandBotset[2]reportedexperimental resultsofgasflowthroughdifferenttypesofporous materials,inwhichtheyobservedthatthegradientofthesquareofthepressurewasproportionaltoapowerofthemass velocity ranging between1 and2 (1 whenthe flow was “completelyviscous”,and 2when it was “completelyturbulent”).

This empirical form ofEqs. (9) and (10)was acceptedto account for inertialflows in porous media over an additional half-centuryduringwhichonlyfewalternativeformswereputforth,likeforinstance[22]

P

L

=

^au

+

^bu2

+

^{cu3 (11)}

witha,b and c being adjustablecoefficients. During the fiftyyears followingForchheimer’s publication, some confusion remained about the physical origin of the quadratic correction to Darcy’s lawas it was often attributed to turbulence, although some references made clearstatements on that point [6].In fact, Irmay [6] argued that there isno reason, in general,toexpectalinearsolutiontothenon-linearNavier–Stokesequations.ThePoiseuillesolutioninstraighttubesisan exceptioncausedbyvanishingcurvature,whichisnotthecaseinrealporousmedia.Agreementhasbeenquiteunanimous on thethresholdvalue ofthe Reynoldsnumberatwhich thecorrection becomes significant,i.e. for1≤^Red≤^{15, where} Red=^ρv_μ^{βdg isbasedonthefiltrationvelocity v}β^{andthetypicalgrainsized}g oftheporousmaterial.However,itwas notbefore 1962andthepublicationby ChauveteauandThirriot[23],inwhichaflow regimeclassification wasproposed, that turbulence in this rangeof Reynolds numbers was dismissed. Turbulence hasbeen confirmed to typically arise for Re_d∼^{100[24–26].}

Duringthisperiod,andevenuptotheendofthe1970s,comparisonswithexperimentalresultswerereported,showing quitegoodagreementforvarioustypesofporousstructures,includingpackedbedsofgrains,bundlesofcapillarytubesor fibrousmediaforflowsofgasesorliquids[27,28].Fromapracticalpointofview,theForchheimermodelhasbeenusedfor applicationsinhydrology,petroleum andchemicalengineering[29,30].From atheoretical pointofview,thesameperiod was marked byvarious attemptsto justifytheform ofEq.(9).Thiswas carried outon thebasis ofdifferent approaches ranging from simplified derivations from the Navier–Stokes equations oranalogies with pressure drop through capillary orifices[6].Theemergenceofmoreformal upscalingtechniquesduring the1970shasledtofurtherdevelopmentsinthe followingyearsthatalsoattemptedtojustifythequadraticformoftheinertialcorrectiontoDarcy’slaw[13,31–33].

2.2. RefinementsontheinertialcorrectiontoDarcy’slaw

Impulsed by the development of both computational resources and numerical methods, the velocity dependence of the correctivetermon Darcy’slaw regainedmuch attentionfromthe early 1990son. Itwas during this periodthat the quadraticvelocitydependenceofthiscorrectionwasquestionedandanalyzedindepth.Numericalsimulationscarriedout

h Thismodel,subjecttotime(

μ

t^∗

ρ

²_β),length-scales(β^r0^{L)andReynoldsnumber ρv}_μ^ββ _L^β

through networks of parallel cylinders of circular cross-section for a flow orthogonal to the cylinders showed that the correction scalesasa3rdpowerofthefiltrationvelocity [34]insteadofaquadraticterm. Thisbehaviorwas theoretically confirmed quasi simultaneously, regardless the geometry at the pore-scale,from formal derivations based on a rigorous upscaling procedure using double-scale homogenizationwith a closureprocess involving periodic representations of the porousmedium[35–37].Theexponent3 was confirmedforaReynoldsnumber, Re_p ,basedonthecharacteristicporesize rangingbetweenδ^1/2 and1,whereδisthemicro-to-macroscaleratio.Thisresultwasfurtheremphasizedlateron[38]and, from its original evidence,led toidentify thisregime asthe so-called“weakinertiaregime”,formalized for homogeneous isotropic andperiodic media.Additional numericalworksoverlargerintervalsoftheReynoldsnumberforflows inmany differentstructures confirmedtheexistence ofweakinertiaandextendedtheclassificationofflowsunder(atleast)three distinct regimes with crossovers, namely [39] i) the weak inertia regime occurring at the onset of non-linearity in the flow-rate-to-pressure-drop relationship forδ^1/2 ≤ ^Rep ≤ ^1; ii) the “stronginertiaregime” characterized by a correction to Darcy’s law that scales asthe square of the filtration velocity, i.e.leading to a Forchheimer type of modelfor Reynolds numbersintherange1 to 10;iii)theturbulentregimeappearingforReynoldsnumberstypicallyoftheorderof100.

Nevertheless, adetailedphysical explanationof acubic orquadratic correctionto Darcy’slaw inorderto account for inertia stillleavesmuchtobedesiredandremainsawidelyopen question[40],whilethedescriptionofthenon-linearity is essentially qualitative. Obviously,inertial macroscopic forces cannot be invoked asthey remain negligible compared to viscous forces[41],andthiscanbeprovedtoholdaslongas Re_p δ⁻¹.Consequently,thenatureofthenon-linearityin therelationshipbetweenthemacroscopicdragforceandfiltrationvelocitymustcertainlybeexplainedfromthesignature ofviscousandinertialforcesatthemicroscale.Ontheonehand,severalmechanismssuggestingthatinertiaalone,atthe pore-scale,canexplainthemacroscalebehaviormaybeputforthsuchas:i)streamlinesbendingduetothetortuosityofthe structure andtolocalconverging-divergingflow patterns;ii) backflowsandseparationsresultingfromformdrag;iii) pore networksactivelyinvolvedintheflowthatarevelocity-dependentasaconsequenceofi)andii) yieldingvariationsinthe dissipation ofthekineticenergy[42].Ontheotherhand,microscaleviscousdrag effectsmaybeconsideredtocontribute to thenon-linearitywhenboundarylayersatthesolid–fluidinterfaces,whichbecomethickerwhentheReynoldsnumber increases(seeanexperimentalobservationin[24]),aretakenintoaccount.Withthismechanism,theinertialcoreflow(in thecenterofthepores)maybeeasilyunderstoodasbeingstronglydependentupontheReynoldsnumber,partlyexplaining thedifferentregimes.

2.3. Furtherdevelopments

Although theexistenceofthe tworegimes(weakandstronginertia)hasbeenwidely accepted,a lotstillneeds tobe understoodregardingtheuniversal existenceanddependenceoftheseregimes(andtheassociatedcrossovers)uponmany parameterssuchasporosity,structuralorder,anisotropy,etc.Inaddition,mostoftheexperimentalornumericalcharacteri- zationsofthemacroscaleinertialcorrectionhavebeencarriedoutin1Dinascalarformalthoughmanyreferencespointed out thattensorialcoefficientsmustbeinvolved[31,36,41,32].Evenifthedevelopmentdidnotallowfortheformalidenti- ficationoftheabove-mentionedregimes,aconvincingderivation,relyingonrigorousupscaling,ofamacroscopicmodelfor momentum transport ofone-phaseflowinginhomogeneous porousmedia withinertiais ^{certainly duetoW itaker[41].}

¹

constraints thatwereclearlyformulated,providesthemacroscalemomentumequation,whichreads

^vβ

= −

^H

μ ^·

∇

^pβ

^β

− ρ

g

(12)

orequivalently

^vβ

= −

^K

μ ^·

∇

^pβ

^β

− ρ

g

−

^F

·

^vβ

⁽¹³⁾

where ^vβ ^{is the filtration (or seepage) velocity as defined in Eq. (8), H is the apparent} permeability, K the intrinsic permeabilityandFtheinertialcorrectiontensor(whichisafunctionofthefiltrationvelocity);∇^pβ^{β isthe}macroscopic pressuregradient,^pβ^{β beingdefinedinEq.(3).Detailsontheaveragingmethodemployedtoobtainthisresultaregiven} in[15].Inadditiontothemacroscopicmodel,theupscaling providesthemeansto determinetheassociatedmacroscopic coefficients (K, H and F contained in Eqs. (12)and (13)) from the solution to the ancillaryproblems (so-called closure problems). Undoubtedly, whilecompletingthephysicaldescription, thiscontributionconcludes someprevious derivations on thesameproblemperformedwithhomogenization[36,40] andopensnewperspectivestomorein-depthinvestigation oftheinertialcorrectiontoDarcy’slaw.

In thisspirit,closure problemswere solved inorder tocompute H andF fordifferentordered anddisordered model structuressoastoinvestigatetheexistenceoftheregimesandtheirdependenceupontheporosityandthemicroscalepore structure[43].Themainimportantconclusionsderivingfromthisanalysiscanbesummarizedasfollows.Thetwotensors HandFaregenerallydenseandnon-symmetricfororderedstructures, evenifthemedium isisotropicatthemacroscale intheDarcyregime.Thenon-symmetrycoincideswithamacroscopicdrag force,whichisnotnecessarilyalignedwiththe macroscopicvelocity.Symmetryisrecoveredforspecificpressuregradientorientationsalongsymmetryaxesofthestructure

(whenpresent).Dissymmetryofthesetensorsdecreaseswhenstructuraldisorderincreases.Whateverthestructure under concern,theweakinertiaregime(acubicinertialcorrectiontoDarcy’slaw)isalwaysobserved.Fororderedstructures,the stronginertiaregime(i.e.a quadratic (Forchheimer)correctiontoDarcy’slaw)doesnotnecessarilyexistanditisotherwise restricted to a narrowinterval of the Reynolds number.For disordered structures, the Forchheimertype ofcorrection is a robustapproximation over avery significant rangeofvaluesof theReynolds number.Moreover, thecrossovervalue of theReynoldsnumberatwhichthequadraticcorrectionbecomesrelevantdecreaseswithincreasingstructuraldisorder,the weakinertiaregimebeingrestrictedtoasmallrangeofReynoldsnumberswherethecorrectionisnotverysignificant.This certainlyexplainswhythisregimeismostofthetimeoverlookedinexperimentalinvestigationsonporousmediahavinga randomporestructure.Inanysituation,usingamodelsuchasthatproposedinEq.(9)impliesthatthepermeabilityinthe linearterminthefiltrationvelocitydiffersfromtheintrinsicpermeability.Evenifsomeprogresshasbeenachievedinthe physicalexplanation andthetheoretical derivationof formalmodels to accountforinertia effects forone-phaseflows in porousmedia,muchworkremainstobe donetounderstandfullythenon-linearitiesassociatedwiththistypeofprocess.

Forinstance,mostoftheanalysessofarwerededicatedtothelaminarsteadyregimeandtheoccurrenceofunsteadiness remains widelyunexplored(thisproblemwasbarely outlinedrecentlyin[44]), aswell asthe turbulentregime;however, furtherdiscussionofthesetopics isoutofthescopeofthepresentreview.

3. Slipflowinporousmedia

Gasflowsinporousmediadifferconsiderablyfromliquid-phaseflows,inparticularforsituationsinwhichtheporesizes arecomparabletothemeanfreepathofthegasmolecules.Thisisthecaseinmanypracticalapplicationsincludingmicro- andnano-fluidicsystemssuchasMEMSandnano-porousmedia,transportinfibrousmedia,gasflowduring soilremedia- tion,long-termnuclearwastedisposal,amongmanyothers.Duetothecurrenthighrelevanceofthistypeoftransport,the evolutionfromthepioneeringworksperformedinthenineteenthcentury tosomeofthecurrentdevelopmentsarebriefly summarizedinthissection.

3.1. Slipflowbackground

Aone-phaseflowinconfinedsystemsofdimensionscomparabletothemeanfree pathleads torarefactioneffectsthat giverisetomanyinterestingcontributionsintransportphenomena.Inhisclassicalstudyofthestressesthatrarefiedgases experience,J.C. Maxwell[45] proposedthat,closetothesurfaceofasolid, thereshouldbeaslidingofthegasincontact with thesolid inthe direction of a tangential stress. Maxwell proposed that the velocity should be proportional to the tangentialstressandinverselyproportionaltotheviscosityofthefluid.Underisothermalconditions,theslidingvelocityfor a1Dflowinthex-directionisexpressedas:

v

=

^Gdv

dx (14)

wherethecoefficientG isthecoefficientofslipping,definedas G

=

²

3

2

f

−

1

λ

β (15)

withλβbeingthemeanfreepathand f thefractionofgasmoleculesthatarediffuselyscatteredatthesurface.Hence,ifthe solidsurfaceiswhollyabsorbent,G=²/3λ_β.TheproposalfromMaxwellisconsistentwithapreviousstudybyNavier[46], andwe shallthus refer to Eq. (14)as theNavier–Maxwell equation. It should be notedthat Maxwell obtained Eq.(14) consideringasingle-componentgas;theextensionofthisequationtomulticomponentmixturesisfoundinJackson[47].

Duringthelastquarterofthenineteenthcentury,rarefactioneffectswereshowntoincreasesignificantlytheflowrate withrespecttothatpredictedfromPoiseuille’slaw[48].Thismotivatedseveralinvestigationsduringtheearlyyearsofthe twentieth century,inparticular thoseby M. Knudsen.The scatteringof gasmoleculesfrom solid wallswas fundamental inKnudsen’stheory [49].Knudsenstudiedmolecularflows intubesanddeterminedthedependenceontubedimensions.

Hediscussed thetransitionfromPoiseuille’sflowintermsoftheratioofthemeanfree pathofthegasmoleculestothe characteristicsizeoftheapparatus(say,β).Thereasonforconsideringthisratioisduetothefactthat,inthecontinuum approach,the slipvelocity maybe understoodasthe averageflow velocity ofthe moleculesata distancefromthe wall that is equalto the meanfree path [50].Hence, asthe meanfree path becomes a bigger fraction ofthe tubediameter, the slip velocity increaseswith respect to the bulk velocity. This important ratio between the mean free path andthe tubediameterisnowadaysknownastheKnudsennumberinhishonor(i.e. K n=λβ/β).Inthecomprehensivereviewby Steckelmacher[51],thehistoricaldevelopmentandrelevanceofKnudsen’sworksarepresentedindetail.

Lateron,Adzumi[52–54],publishedaseriesofpapersdedicatedtothestudyofgasflowthroughcapillaries.Heconsid- eredthreecases:1) whenλβ isverysmallincomparisontothediameterofthecapillary(i.e. K n^{1);2) when}λβ islarge comparedto thediameter(i.e. K n>1), and3) whenλβ iscomparabletothecapillarydiameter(i.e. K n∼^{1).Inthefirst} case, hefound Poiseuille’slawto bequitesuitable, andthe flowisinverselyproportionalto thegasviscosity[52].Inthe secondcase,theflowratewasfoundtobeindependentoftheviscosity,butinverselyproportionaltothesquarerootofthe

gasmolecularweight,M, [53].TheflowcharacteristicsinthethirdcasewerefoundbyAdzumitobeacombinationofthe twofirstones[54].Nowadays,thereissomeconsensusthatthefollowingboundsareidentifiable:forK n<10⁻³,thelaws ofcontinuummechanicsaresafelyapplicable,andnon-slipcanbeassumedatthesolidboundaries;for10⁻³ <K n<10⁻¹, theflowregimecorrespondstoslipflow,andtheNavier–Maxwellequationmustbeconsideredatthesolid–fluidinterface;

for10⁻¹ <K n<10,thereisa transitionregime inwhichthelawsofcontinuum mechanicsare likelytofailbecausethe continuumhypothesisisnolongersatisfied.Forporousmediaapplicationsinthetransitionregime,Maxwell[55]proposed that theactionoftheporousmaterialoverthegaswassimilartoanumberofdustparticlesofthemoving system,hence givingrisetothewell-known dustygasmodel.Thismodelhasthenicefeaturethattheinteractions ofgasmoleculeswith the dust molecules simulatetheir interaction withthe rigid porousmatrix, thus avoidingthe problemof flux variations acrossthesectionsofthepores [47].Finally,forK n>10 thegaskinetictheorymustbeconsideredbecausedescriptionin termsofparticle–wallcollisionoperatorsisrequired.ThislimitingsituationiscalledmolecularstreamingorKnudsenflow, anditischaracterizedbythefactthattheflowtakesplacebydiffusion,insteadofviscous,mechanisms[50].Here,theterm diffusionmeansthattheflowresultsfromcreepatthewallratherthanfrommolecule-to-moleculecollisions.

3.2. Pore-scaleslipflowanditsconsequenceonDarcy’slaw

Few years after Adzumi’s works, in his study of the permeability of gas flows in porous media in the slip regime, Klinkenberg [19] found that this coefficient isalmost a linear function ofthe reciprocalmean pressure, ^pβ^{β.Using an} idealizedporousmediumrepresentationconsistinginanarrayofcapillaries,Klinkenbergwasabletodeducethefollowing relationbetweentheapparentpermeability,K_s ,andtheintrinsicpermeability,K:

K_s

=

^K

1

+

^b

^pβ

^β

(16) withbbeingasurface- andgas-dependentconstant.Evidently,atsufficientlylargegaspressures,KsapproachesK.Klinken- bergusedtheaboveequationtopredictthevaluesofK_s/K indifferentexperimentalconditionswithreliableaccuracy,thus showing the need to consider slip effects in the determination of the permeability. In this way, the Darcy–Klinkenberg modelis(gravityisomittedhere):

^vβ

= −

^K

μ

1

+

^b

^pβ

^β

∂

^pβ

^β

∂

x (17)

foranaverageone-dimensionalflowinthex-direction,while

μ

isthefluiddynamicviscosity.

Animportantpointofdiscussionintherecentliteraturerelatedtogastransportinporousmediaisaboutthepertinence ofthelinearandfirst-orderinterfacial boundaryconditiongiveninEq.(14).Shenet al.[56] derivedafirst-orderslipcon- ditionfromtheChapman–EnskogsolutiontotheBoltzmannequation.ThisapproachconsistsinlinearizingtheBoltzmann equation usingaperturbationexpansionfortheprobabilityfunctionintermsoftheKnudsennumber.Theresultingcondi- tion includesanadditionaltermduetothepressuregradient alongtheflow’sdirection. Thesuccessoffirst-ordermodels hasbeenarguedtobe constrainedtoslightlyrarefiedgasflowsbyDeissler[57].Accordingtothisauthor,asthepressure inthegasbecomessmaller,thevelocityprofilesmaybenonlinearoveradistancefromthesolidsurfacecorrespondingto themeanfreepathandthejumpsattheinterfacemaybeexpectedtobefunctionsofhigher-ordernormalandtangential derivatives.Deisslerthusproposedtouseasecond-orderboundaryconditionthatmatchestheNavier–Stokesequationsfor slipflows,finding goodagreementwithexperimental results.Anotherearlyproposalofsecond-ordermodels isthescalar one byCercignani[58] onthebasisoftheBhatnagar–Gross–Krook(BGK)approximationoftheBoltzmannequation.How- ever, theseextensionsconsist ofmodificationsto thescalarequation (14),inother words,theyare constrainedtosimple geometries whereone-dimensional flow is applicable. Unfortunately, extensionsto more complicatedgeometries are not straightforwardandtheyremainachallenge.Furthermore,sincetheNavier–Stokesequationsarefirst-orderaccurateinthe Knudsennumber,itisnoteasytojustifytheuseofhigher-orderboundaryconditions.Hence,analternativeistousehigher ordermomentumtransportequations,suchastheBurnettequation [59],whicharetheresultofkeepingthesecond-order termsintheChapman–EnskogapproachtoapproximatetheBoltzmannequation.

As mentioned above, one ofthe limitations ofthe Navier–Maxwellboundary condition is that it isrestricted to one- dimensionalflows.ThislimitationisalsosharedbytheKlinkenbergmodel.Toaddressthisissue,Einzelet al.[60]proposed ageneralizedversionoftheslipboundarycondition,whichcanbeexpressedasfollows:

vβ

= −

2

− σ

_ν

σ

_ν

ξ

λ

βn

·

∇

vβ

+ ∇

v^T_β

· (

I

−

nn

)

(18)

where

σ

ν isthetangential-momentumaccommodationcoefficient, nistheunitnormalvector directedfromthe fluidto- wardthesoldphaseandIistheidentitytensor.Thecoefficient

σ

ν accountsfortheaveragetangentialmomentumexchange betweenthemoleculesandthefluid,andcanvaryfromzerotoone.

3.3. Theoreticalmacroscopicslipflowmodelsinporousmedia

In the same lineof thoughts as those mentioned for inertialflows, theKlinkenberg model, although widely used in theliterature,wasformallyderived onlymorethanhalfacenturyafteritspublication.Usingthehomogenizationmethod, SkjetneandAuriault [61] considered thesteady-state,low-velocity Navier–Stokesequations forcompressibleflows inthe slipregime,i.e.Re K n 1 (for Re= ^O(δ = /L),withandLbeingthecharacteristiclengthscalesatthemicroscaleand macroscale,respectively).Inthiswork,forthefirsttime,avectorialformoftheKlinkenbergmodelwasrigorouslydeduced forconditionsinwhich bothlocalcompressibilityandinertial effectsaresmaller thanthe wall-slipeffects.The apparent permeabilitytensorwasfoundtobepositive-definiteandnon-symmetric,ingeneral.Thisstudywassubsequentlyexpanded by Chastanet et al.[62] to derive the corresponding upscaledmodel for low-pressure gas flows in dual-porosity media includingfractures.TheirstudyhighlightedthattheKnudsennumbershouldbeconsideredinadditiontotheseparationof characteristiclengthscalesinthesysteminordertoassessthedomainsofvalidityofupscaledmodelsforgasflow.

The derivationof theeffective-medium equation corresponding toslightly-compressible slip-flowconditions usingthe volumeaveragingmethodhasbeencarriedoutrecentlybyLasseuxetal.[63].Thisworkcompletesthepreviousderivations bySkjetneandAuriaultonthefollowingpoints:1)thecompressibilityeffectsare takenintoaccountintheframeworkof slightlycompressibleflowsrestrictedtosmallReynoldsnumbersandsmallfrequencynumber;2) thevectorialformofthe slipboundaryconditionisconsideredinitscompleteform,i.e.includingthecompleteshear-rateatthefluid-solidinterface asshowninEq.(18);3)itisderivedforabarotropicfluidwithoutanyassumptionontheequationofstateofthegas.For anidealgas,theresultingupscaledmodelis

^vβ

= −

¹

μ

^K

^·

⎛

⎝

I

+ ξ μ

^pβ

^β

π

^R

^Tβ

^β

2M S

⎞

⎠

(19)

whichinvolvestwotensors,namely,theintrinsicpermeabilitytensor,K,andaslip-flowcorrectiontensor,S.Theparallelism betweenEqs.(17)and(19)isobvious.Theancillaryclosureproblemrequiredtopredictthevaluesoftheeffective-medium coefficients was derived andformally solved for simple porous medium geometries in two- and three-dimensional unit cells. Furthermore,the dependence of s (S=^{sI) on K (K}=^{KI) was found to obey a power-law} relationship, with the value ofthe exponentdepending on the geometricalconfiguration. Ina subsequentwork by Lasseux et al.[64],the slip correction was moreaccurately described by considering an expansion inthe Knudsen numberatthe closurelevel.This leadstoareformulationoftheclosureproblemasadifferential(insteadofanintegro-differential)boundary-valueproblem, whichwas solved inmorecomplicatedunit cellsin ordertopredict theapparent permeabilitytensor.Furthermore,with thisexpansion,theslip-flowcorrectiontensorwas showntobethesumofslipcorrectionsatthesuccessiveordersofthe Knudsennumber.Theconsiderationofthecompleteformoftheboundaryconditionatthesolid–fluidinterface wasfound to becrucialfor thepredictionofthe slipcorrectionsatthedifferentorders of K n.Theiranalysisevidenced anonlinear relationship betweenthe apparent permeabilityandthe Knudsen number.This relationship motivatesfurther theoretical andexperimentalresearchonthesubject,inparticularforhighlyporousstructures.

4. Conclusions

Thiswork hasbeen dedicatedto the analysisofthe evolutionof two majormodifications to Darcy’slaw, namelythe inclusion of inertial and slipeffects. The review carried out for both extensions suggests the following conclusions and prospects.

– TheempiricalintroductionofaquadraticcorrectionintermsofthefiltrationvelocitytoDarcy’slawbyForchheimerin 1901for1Dflowswascertainlyinspiredbyamodelthatwascommonlyusedforflowsinpipespriortothepublication ofDarcy’slaw.Thiscorrectionhasbeenwidelysupportedbyempiricismformorethan90years,aperiodafterwhicha firsttheoretical3Dmodelwasachievedby upscaling(homogenization),showingthattheonsetofdeviationfromDar- cy’slawduetoinertiainvolvesacorrectionthatratherscalesasa3rdpowerofthefiltrationvelocityinaweak-inertia regime.ForlargerReynoldsnumbervalues,aso-calledstronginertiaregime,wheretheForchheimercorrectionisdue tohold,wasaccepted.Anupscaled3Dcompletemodel,obtainedbyvolumeaveragingfiveyearslater,wasusedtohigh- lightthefact that,iftheweakinertiaregime always exists,thequadraticcorrectiondoesnotholdinsomeparticular situationsofpore-scaleorderedstructuresandthat,however,inthepresenceofdisorder,theForchheimer-typecorrec- tionisarobustone.Atthispoint,aquitedifferentobservationfromthatindicatedbelowforslipflowsmaybepointed out regardingcorrectionsmadetoDarcy’slawinorderto includeinertialeffects.Indeed,iftheunderlyingphysics at theporescaledoesnotriseanyparticularquestion,issuesaremainlyrelatedtotheunderstandingofthedifferentflow regimes atthe macroscale, the physicalmechanisms that trigger thetransition fromone regime to another andthe associatedrangeofReynoldsnumbers,togetherwiththepossibleoccurrenceofunsteadiness.

– Theoriginalidentificationoftheslipeffectforgasflowsinporousmedia,whentheKnudsennumberisnotexceedingly smallcomparedtounity,by Klinkenberg(1941),whichledthisauthortoproposeacorrectiontotheintrinsicperme- ability(inthe1Dcase)inverselyproportionaltothemeanpressure,hasbeenacceptedwithempiricaljustificationsfor almost60years.Afterthisperiod,afirsttheoretical3Dmodelwasderived,followedbymorerefinedones15yearslater

using rigorousupscaling techniques.While theseupscaling toolsare nowathandfor suchtheoretical developments, it clearlyappears that themainissueslie inan appropriate pore-scaledescriptionofthephysics inthat case,andin particular, intheslip-boundary condition, whichstill requiressome effortstobettercapture, andpossiblyextend, its domainofapplicationintermsoftheKnudsennumber.Inthesamespirit,despitesomeattempts,macroscalemodels fortransitionalandstronglyrarefiedflowregimesstillrequireimportantefforts.Inparallel,numericalandexperimental investigationsarenecessarytohighlighttheunderstandingofthedetailedphysics.

As aprospect,onemayfinallyconcludewiththefollowing.FromtheevolutionoftwomodificationstoDarcy’slawinves- tigated inthepresentwork,itisimportanttoremarkthat,afterone-dimensional correctionswere proposed,followedby extensiveexperimentalanalyses,therewasalongtimeperiodbeforerigorousdeductionswerepresented.Thisdelaycanbe explainedbythefactthatthedifferentnecessarytheoreticalframeworksforperformingupscalingfromthepore-scaletothe macroscalearerelativelynew(about30–40yearsold)andalsobythefactthat,inmanysituations,theone-dimensionalver- sionsremainedrelativelysatisfactory.However,withrecentapplicationsdirectedtomicro- andnanofluidicdevices,among others,there isa needformoreaccurate macroscalemodels,that can bevalidatedthrough comparisonwithreliableex- perimentaldataanddirectnumericalsimulations.Inthislastissue,thereisastrongtendencyincurrentresearchtowards understanding macroscalephenomenathroughimaginganddirectsimulations.Thishasbeenmadepossibleby recentad- vances incomputationalcapabilities. Withthisperspective inmind,itisrelevanttoposethequestion ofwhichdirection future advancesofflows inporous mediawill take.Probably, experimental andnumerical studieswouldstill evolvevery significantly.However,strongeffortsshouldcertainlybededicatedtomoresophisticatedupscalingapproachesapplicableto morechallengingsituationsthanthosestudiedinthecurrentstateoftheart.

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